What are the “real math” connections between Euclidean Geometry and Complex Numbers?

manpreet Best Answer 2 years ago

Some background: I am a high school student and I am very interested in math. I have done a lot of the extracurricular learning I have done is through doing math problems from various competitions, whether that be the Crux Mathematicorum or the AIME. In doing these problems, I have found various ways that complex numbers can be used to help solve a geometry problem, and vice versa. For example, expressing coordinates as complex numbers and rotating them by multiplying by ekπekπ for some rational number kk is something I have seen commonly, and I have seen a derivation of Heron's formula using complex numbers.

However, something I would like to know about is the connection between the two in the "real" modern math world. Are the two fields deeply and fundamentally intertwined, or do they just provide each other with tools that can be used to aid each other?